trigonometry

Geometry and trigonometry
15
Trigonometric waveforms
15.1 Graphs of trigonometric functions
By drawing up tables of values from 0 ◦ to 360 ◦ ,
graphs of y = sin A, y = cos A and y = tan A may be
plotted. Values obtained with a calculator (correct
to 3 decimal places—which is more than sufficient
for plotting graphs), using 30 ◦ intervals, are shown
below, with the respective graphs shown in Fig. 15.1.
(a) y = sin A
A 0 30 ◦ 60 ◦ 90 ◦ 120 ◦ 150 ◦ 180 ◦
sin A 0 0.500 0.866 1.000 0.866 0.500 0
A 210 ◦ 240 ◦ 270 ◦ 300 ◦ 330 ◦ 360 ◦
sin A −0.500 −0.866 −1.000 −0.866 −0.500 0
(b) y = cos A
A 0 30 ◦ 60 ◦ 90 ◦ 120 ◦ 150 ◦ 180 ◦
cos A 1.000 0.866 0.500 0 −0.500 −0.866 −1.000
A 210 ◦ 240 ◦ 270 ◦ 300 ◦ 330 ◦ 360 ◦
cos A −0.866 −0.500 0 0.500 0.866 1.000
(c) y = tan A
A 0 30 ◦ 60 ◦ 90 ◦ 120 ◦ 150 ◦ 180 ◦
tan A 0 0.577 1.732 ∞ −1.732 −0.577 0
A 210 ◦ 240 ◦ 270 ◦ 300 ◦ 330 ◦ 360 ◦
tan A 0.577 1.732 ∞ −1.732 −0.577 0
From Figure 15.1 it is seen that:
(i) Sine and cosine graphs oscillate between peak
values of ±1.
(ii) The cosine curve is the same shape as the sine
curve but displaced by 90 ◦ .
Figure 15.1
(iii) The sine and cosine curves are continuous and
they repeat at intervals of 360 ◦ ; the tangent
curve appears to be discontinuous and repeats
at intervals of 180 ◦ .
15.2 Angles of any magnitude
(i) Figure 15.2 shows rectangular axes XX ′ and YY ′
intersecting at origin 0.As with graphical work,
measurements made to the right and above 0 are
positive while those to the left and downwards
are negative. Let OA be free to rotate about 0.